As a keen shootist I have had one or two bird on the kitchen table and also kept a few feathers.
Most birds are very clean indeed, some have a few mites and fleas. The only time I have seen infestation on a bird is when the bird was ill.
As soon as the bird dies the bugs are off looking for a new home so you have more in the game bag than in the kitchen.
Problem is with all this traffic the ones jumping ship may find your tying stock and attempt to set up home. Not likely they will stay but who wants a desk full of homeless bugs.
I say feathers only, a couple of hours in the freezer. Wings etc, a day and then a good dry out before storage.
And if you keep bringing warm uns into the house for plucking etc keep them away from your tying desk.